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Thread: Isomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    Isomorphism

    Define for each $\displaystyle \lambda\in\mathbb{R}$ a natural action on $\displaystyle T_{\alpha}$ by rotation:
    $\displaystyle
    \tau(\lambda)(f)(t) = \left\{
    \begin{array}{lr}
    f(t+\lambda) & : t+\lambda \in [0,1]\\
    \alpha^n(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z}
    \end{array}
    \right.
    $
    where $\displaystyle \alpha$ is an automorphism of the $\displaystyle C^*$-algebra A. Prove that the one parameter family $\displaystyle \lambda\mapsto \tau(\lambda)$ gives rise to an isomorphism

    $\displaystyle
    T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))
    $

    where $\displaystyle T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}$
    Last edited by Mauritzvdworm; Sep 27th 2010 at 11:42 AM. Reason: typing error
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    Define for each $\displaystyle \lambda\in\mathbb{R}$ a natural action on $\displaystyle T_{\alpha}$ by rotation:
    $\displaystyle
    \tau(\lambda)(f)(t) = \left\{
    \begin{array}{lr}
    f(t+\lambda) & : t+\lambda \in [0,1]\\
    \alpha(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z}
    \end{array}
    \right.
    $
    where $\displaystyle \alpha$ is an automorphism of the $\displaystyle C^*$-algebra A. Prove that the one parameter family $\displaystyle \lambda\mapsto \tau(\lambda)$ gives rise to an isomorphism

    $\displaystyle
    T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))
    $

    where $\displaystyle T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}$
    I can't help with this problem, which probably requires a fairly extensive knowledge of semidirect products (and in any case I am about to go away for a week). But the problem as stated has two obvious mistakes.

    First, the definition of $\displaystyle \tau(\lambda)$ is wrong. It does not define an automorphism group. In fact, the definition of $\displaystyle \tau(\lambda)(f)(t)$ does not even define a continuous function of t at integer values of $\displaystyle t+\lambda$. The correct definition should presumably be $\displaystyle \tau(\lambda)(f)(t) = \alpha^n(f(t+\lambda-n))$ when $\displaystyle t+\lambda\in[n,n+1]$.

    Second, the semidirect product $\displaystyle T_\alpha\rtimes_\alpha\mathbb{R}$ does not make sense. It should be $\displaystyle T_\alpha\rtimes_\lambda\mathbb{R}.$
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